Ambiguous Triangles [closed]

How can I determine whether an ambiguous triangle has has one answer, two answers, or none? I understand that an ambiguous triangle is that which presents two sides and one angle (SSA), I just don't understand how to know whether it has one answer, or two.

closed as unclear what you're asking by Adam Hughes, Matt Samuel, Chappers, Daniel, graydadMay 20 '15 at 0:52

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Suppose that $A$ is the side adjacent to the angle, and $B$ is not. Let $A$ have length $a$ and $B$ have length $b$, and let the angle be $\theta$.

If $b > a$, then there is $1$ solution for all values of $\theta$

Assuming $\theta < \frac{\pi}{2}$ and $b \leq a$:

0 solutions: $$b < a\sin(\theta)$$ 1 solution: $$b = a\sin(\theta)$$ 2 solutions: $$b > a\sin(\theta)$$

If $\theta = \frac{\pi}{2}$, then there is a degenerate solution where $a=b$.

If $\theta > \frac{\pi}{2}$, then there are no solutions where $a < b$, and a degenerate solution where $a=b$.

I should add that a good intuition for this behavior comes from trying to construct such a triangle with a compass and straightedge, starting with the angle, then edge A, then edge B.

• It might be worth considering cases when $b \gt a$, or when $b=a$, or when $\theta$ is bigger than a right angle and $a\sin(\theta) \le b \lt a$. – Henry May 19 '15 at 20:25
• This is not quite right. For example, if $b>a$, then there is exactly one solution. – Jack Lee May 19 '15 at 20:31
• One should probably stipulate that $\theta < \frac\pi2$ and $b < a$; otherwise I don't think there can be more than one non-degenerate solution (if there is even one) for which the desired SSA properties hold. (I assume $\theta$ is an interior angle by the definition of SSA.) – David K May 19 '15 at 20:33
• There are only single or degenerate solutions in the other cases, though I shouldn't have neglected them (added now). – TokenToucan May 19 '15 at 20:38
• If $b\gt a$, there is one solution only if we assume that the triangle has non-negative angles and sides. – John Joy May 19 '15 at 21:31